Lieb's square ice constant
| Representations | |
|---|---|
| Decimal | 1.53960071783900203869106341467188… |
| Algebraic form | |
For the number of possible Eulerian orientations of an n × n periodic grid graph denoted , Lieb's square ice constant is the limit of as approaches infinity.
![{\displaystyle {\sqrt[{n^{2}}]{f(n)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba02db2aa8496e27cb040de2a33791ef6af1511)
Lieb's square ice constant is a mathematical constant used in the field of combinatorics to quantify the number of Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967.[1]
Definition
An n × n grid graph (with periodic boundary conditions and n ≥ 2) has n2 vertices and 2n2 edges; it is 4-regular, meaning that each vertex has exactly four neighbors. An orientation of this graph is an assignment of a direction to each edge; it is an Eulerian orientation if it gives each vertex exactly two incoming edges and exactly two outgoing edges.
Denote the number of Eulerian orientations of this graph by f(n). Then
![{\displaystyle \lim _{n\to \infty }{\sqrt[{n^{2}}]{f(n)}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8{\sqrt {3}}}{9}}=1.5396007\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/146be733277faf0518d5db191a4a010dee25e4d9)
is Lieb's square ice constant. Lieb used a transfer-matrix method to compute this exactly.
The function f(n) also counts the number of 3-colorings of grid graphs, the number of nowhere-zero 3-flows in 4-regular graphs, and the number of local flat foldings of the Miura fold.[3] Some historical and physical background can be found in the article Ice-type model.
See also
References
- ^ Lieb, Elliott (1967). "Residual Entropy of Square Ice". Physical Review. 162 (1): 162. Bibcode:1967PhRv..162..162L. doi:10.1103/PhysRev.162.162.
- ^ (sequence A118273 in the OEIS)
- ^ Ballinger, Brad; Damian, Mirela; Eppstein, David; Flatland, Robin; Ginepro, Jessica; Hull, Thomas (2015), "Minimum forcing sets for Miura folding patterns", Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp. 136–147, arXiv:1410.2231, doi:10.1137/1.9781611973730.11, ISBN 978-1-61197-374-7, S2CID 10478192